Precessing vortex cores (PVC), arising from a global instability in swirling flows, can dramatically alter the dynamics of swirl-stabilized flames. Previous study of these instabilities has identified their frequencies and potential for interaction with the shear layer instabilities also present in swirling flows. In this work, we investigate the dynamics of precessing vortex cores at a range of swirl numbers and the impact that turbulence, which tends to increase with swirl number due to the increase in mean shear, has on the dynamics of this instability. This is particularly interesting as stability predictions have previously incorporated turbulence effects using an eddy viscosity model, which only captures the impact of turbulence on the base flow, not on the instantaneous dynamics of the PVC itself. Time-resolved experimental measurements of the three-component velocity field at ten swirl numbers show that at lower swirl numbers, the PVC is affected by turbulence through the presence of vortex jitter. With increasing swirl number, the PVC jitter decreases as the PVC strength increases. There is a critical swirl number below which jitter of the PVC vortex monotonically increases with increasing swirl number, and beyond which the jitter decreases, indicating that the strength of the PVC dominates over turbulent fluctuations at higher swirl numbers, despite the fact that the turbulence intensities continue to rise with increasing swirl number. Further, we use a nonlinear van der Pol oscillator model to explain the competition between the random turbulent fluctuations and coherent oscillations of the PVC. The results of this work indicate that while both the strength of the PVC and magnitude of turbulence intensity increase with increasing swirl number, there are defined regimes where each of them hold a stronger influence on the large-scale, coherent dynamics of the flow field.